The number of orbitals associated with quantum numbers `n =4 and m _(8) =-1/2 ` are

Options

A. 16
B. 32
C. 8
D. 4

Explanation

To solve the problem of finding the number of orbitals associated with the quantum numbers \( n = 4 \) and \( m_s = -\frac{1}{2} \), we can follow these steps:

### Step 1: Understand the Quantum Numbers
The quantum numbers describe the properties of electrons in an atom. The principal quantum number \( n \) indicates the energy level or shell, while the spin quantum number \( m_s \) indicates the direction of the electron's spin.

### Step 2: Determine the Value of \( l \)
For a given principal quantum number \( n \), the azimuthal quantum number \( l \) can take values from \( 0 \) to \( n-1 \). Therefore, for \( n = 4 \):
- Possible values of \( l \) are \( 0, 1, 2, 3 \).

### Step 3: Identify the Subshells
Each value of \( l \) corresponds to a specific subshell:
- \( l = 0 \) corresponds to the \( s \) subshell (1 orbital)
- \( l = 1 \) corresponds to the \( p \) subshell (3 orbitals)
- \( l = 2 \) corresponds to the \( d \) subshell (5 orbitals)
- \( l = 3 \) corresponds to the \( f \) subshell (7 orbitals)

### Step 4: Count the Total Number of Orbitals
Now, we can count the total number of orbitals for each subshell:
- For \( l = 0 \) (s): 1 orbital
- For \( l = 1 \) (p): 3 orbitals
- For \( l = 2 \) (d): 5 orbitals
- For \( l = 3 \) (f): 7 orbitals

Adding these together:
\[ 1 + 3 + 5 + 7 = 16 \text{ orbitals} \]

### Step 5: Consider the Spin Quantum Number
The spin quantum number \( m_s \) can take values of \( +\frac{1}{2} \) or \( -\frac{1}{2} \). Each orbital can hold two electrons with opposite spins. Since we are asked for the orbitals associated with \( m_s = -\frac{1}{2} \), we still have the same number of orbitals, which is 16.

### Final Answer
Thus, the total number of orbitals associated with quantum numbers \( n = 4 \) and \( m_s = -\frac{1}{2} \) is **16**.

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